metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.S3⋊1C4, D4.1(C4×S3), C6.31(C4×D4), C4⋊C4.129D6, Dic6⋊1(C2×C4), C24⋊C4⋊15C2, C6.Q16⋊2C2, (C2×C8).165D6, D4⋊C4.7S3, (C2×D4).124D6, C2.1(D8⋊S3), C12.2(C22×C4), (D4×Dic3).1C2, C22.66(S3×D4), Dic6⋊C4⋊1C2, C3⋊2(SD16⋊C4), C6.24(C8⋊C22), C2.Dic12⋊19C2, C2.1(D4.D6), (C6×D4).18C22, C12.143(C4○D4), C4.44(D4⋊2S3), (C2×C12).197C23, (C2×C24).219C22, (C2×Dic3).133D4, C6.24(C8.C22), C4⋊Dic3.57C22, (C4×Dic3).3C22, (C2×Dic6).49C22, C2.15(Dic3⋊4D4), C3⋊C8⋊1(C2×C4), C4.2(S3×C2×C4), (C3×D4).1(C2×C4), (C2×C3⋊C8).5C22, (C2×C6).210(C2×D4), (C3×C4⋊C4).2C22, (C2×D4.S3).1C2, (C3×D4⋊C4).10C2, (C2×C4).304(C22×S3), SmallGroup(192,316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.S3⋊C4
G = < a,b,c,d,e | a4=b2=c3=e4=1, d2=a2, bab=dad-1=eae-1=a-1, ac=ca, bc=cb, dbd-1=ebe-1=ab, dcd-1=c-1, ce=ec, ede-1=a2d >
Subgroups: 312 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, C22×Dic3, C6×D4, SD16⋊C4, C6.Q16, C24⋊C4, C2.Dic12, C3×D4⋊C4, Dic6⋊C4, C2×D4.S3, D4×Dic3, D4.S3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, D4⋊2S3, SD16⋊C4, Dic3⋊4D4, D8⋊S3, D4.D6, D4.S3⋊C4
►On 96 points
Generators in S96
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 31)(33 35)(38 40)(41 43)(45 47)(50 52)(53 54)(55 56)(57 58)(59 60)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 79)(81 83)(86 88)(89 91)(93 95)
(1 6 17)(2 7 18)(3 8 19)(4 5 20)(9 28 22)(10 25 23)(11 26 24)(12 27 21)(13 90 94)(14 91 95)(15 92 96)(16 89 93)(29 38 35)(30 39 36)(31 40 33)(32 37 34)(41 52 45)(42 49 46)(43 50 47)(44 51 48)(53 70 76)(54 71 73)(55 72 74)(56 69 75)(57 68 62)(58 65 63)(59 66 64)(60 67 61)(77 83 86)(78 84 87)(79 81 88)(80 82 85)
(1 91 3 89)(2 90 4 92)(5 15 7 13)(6 14 8 16)(9 82 11 84)(10 81 12 83)(17 95 19 93)(18 94 20 96)(21 86 23 88)(22 85 24 87)(25 79 27 77)(26 78 28 80)(29 61 31 63)(30 64 32 62)(33 58 35 60)(34 57 36 59)(37 68 39 66)(38 67 40 65)(41 56 43 54)(42 55 44 53)(45 69 47 71)(46 72 48 70)(49 74 51 76)(50 73 52 75)
(1 41 10 35)(2 44 11 34)(3 43 12 33)(4 42 9 36)(5 49 28 30)(6 52 25 29)(7 51 26 32)(8 50 27 31)(13 74 78 64)(14 73 79 63)(15 76 80 62)(16 75 77 61)(17 45 23 38)(18 48 24 37)(19 47 21 40)(20 46 22 39)(53 82 57 92)(54 81 58 91)(55 84 59 90)(56 83 60 89)(65 95 71 88)(66 94 72 87)(67 93 69 86)(68 96 70 85)
G:=sub<Sym(96)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(33,35)(38,40)(41,43)(45,47)(50,52)(53,54)(55,56)(57,58)(59,60)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,79)(81,83)(86,88)(89,91)(93,95), (1,6,17)(2,7,18)(3,8,19)(4,5,20)(9,28,22)(10,25,23)(11,26,24)(12,27,21)(13,90,94)(14,91,95)(15,92,96)(16,89,93)(29,38,35)(30,39,36)(31,40,33)(32,37,34)(41,52,45)(42,49,46)(43,50,47)(44,51,48)(53,70,76)(54,71,73)(55,72,74)(56,69,75)(57,68,62)(58,65,63)(59,66,64)(60,67,61)(77,83,86)(78,84,87)(79,81,88)(80,82,85), (1,91,3,89)(2,90,4,92)(5,15,7,13)(6,14,8,16)(9,82,11,84)(10,81,12,83)(17,95,19,93)(18,94,20,96)(21,86,23,88)(22,85,24,87)(25,79,27,77)(26,78,28,80)(29,61,31,63)(30,64,32,62)(33,58,35,60)(34,57,36,59)(37,68,39,66)(38,67,40,65)(41,56,43,54)(42,55,44,53)(45,69,47,71)(46,72,48,70)(49,74,51,76)(50,73,52,75), (1,41,10,35)(2,44,11,34)(3,43,12,33)(4,42,9,36)(5,49,28,30)(6,52,25,29)(7,51,26,32)(8,50,27,31)(13,74,78,64)(14,73,79,63)(15,76,80,62)(16,75,77,61)(17,45,23,38)(18,48,24,37)(19,47,21,40)(20,46,22,39)(53,82,57,92)(54,81,58,91)(55,84,59,90)(56,83,60,89)(65,95,71,88)(66,94,72,87)(67,93,69,86)(68,96,70,85)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(33,35)(38,40)(41,43)(45,47)(50,52)(53,54)(55,56)(57,58)(59,60)(61,64)(62,63)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,79)(81,83)(86,88)(89,91)(93,95), (1,6,17)(2,7,18)(3,8,19)(4,5,20)(9,28,22)(10,25,23)(11,26,24)(12,27,21)(13,90,94)(14,91,95)(15,92,96)(16,89,93)(29,38,35)(30,39,36)(31,40,33)(32,37,34)(41,52,45)(42,49,46)(43,50,47)(44,51,48)(53,70,76)(54,71,73)(55,72,74)(56,69,75)(57,68,62)(58,65,63)(59,66,64)(60,67,61)(77,83,86)(78,84,87)(79,81,88)(80,82,85), (1,91,3,89)(2,90,4,92)(5,15,7,13)(6,14,8,16)(9,82,11,84)(10,81,12,83)(17,95,19,93)(18,94,20,96)(21,86,23,88)(22,85,24,87)(25,79,27,77)(26,78,28,80)(29,61,31,63)(30,64,32,62)(33,58,35,60)(34,57,36,59)(37,68,39,66)(38,67,40,65)(41,56,43,54)(42,55,44,53)(45,69,47,71)(46,72,48,70)(49,74,51,76)(50,73,52,75), (1,41,10,35)(2,44,11,34)(3,43,12,33)(4,42,9,36)(5,49,28,30)(6,52,25,29)(7,51,26,32)(8,50,27,31)(13,74,78,64)(14,73,79,63)(15,76,80,62)(16,75,77,61)(17,45,23,38)(18,48,24,37)(19,47,21,40)(20,46,22,39)(53,82,57,92)(54,81,58,91)(55,84,59,90)(56,83,60,89)(65,95,71,88)(66,94,72,87)(67,93,69,86)(68,96,70,85) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(14,16),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,31),(33,35),(38,40),(41,43),(45,47),(50,52),(53,54),(55,56),(57,58),(59,60),(61,64),(62,63),(65,68),(66,67),(69,72),(70,71),(73,76),(74,75),(77,79),(81,83),(86,88),(89,91),(93,95)], [(1,6,17),(2,7,18),(3,8,19),(4,5,20),(9,28,22),(10,25,23),(11,26,24),(12,27,21),(13,90,94),(14,91,95),(15,92,96),(16,89,93),(29,38,35),(30,39,36),(31,40,33),(32,37,34),(41,52,45),(42,49,46),(43,50,47),(44,51,48),(53,70,76),(54,71,73),(55,72,74),(56,69,75),(57,68,62),(58,65,63),(59,66,64),(60,67,61),(77,83,86),(78,84,87),(79,81,88),(80,82,85)], [(1,91,3,89),(2,90,4,92),(5,15,7,13),(6,14,8,16),(9,82,11,84),(10,81,12,83),(17,95,19,93),(18,94,20,96),(21,86,23,88),(22,85,24,87),(25,79,27,77),(26,78,28,80),(29,61,31,63),(30,64,32,62),(33,58,35,60),(34,57,36,59),(37,68,39,66),(38,67,40,65),(41,56,43,54),(42,55,44,53),(45,69,47,71),(46,72,48,70),(49,74,51,76),(50,73,52,75)], [(1,41,10,35),(2,44,11,34),(3,43,12,33),(4,42,9,36),(5,49,28,30),(6,52,25,29),(7,51,26,32),(8,50,27,31),(13,74,78,64),(14,73,79,63),(15,76,80,62),(16,75,77,61),(17,45,23,38),(18,48,24,37),(19,47,21,40),(20,46,22,39),(53,82,57,92),(54,81,58,91),(55,84,59,90),(56,83,60,89),(65,95,71,88),(66,94,72,87),(67,93,69,86),(68,96,70,85)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | C8⋊C22 | C8.C22 | D4⋊2S3 | S3×D4 | D8⋊S3 | D4.D6 |
| kernel | D4.S3⋊C4 | C6.Q16 | C24⋊C4 | C2.Dic12 | C3×D4⋊C4 | Dic6⋊C4 | C2×D4.S3 | D4×Dic3 | D4.S3 | D4⋊C4 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D4.S3⋊C4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 72 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 62 | 60 | 0 | 0 | 0 | 0 |
| 71 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 24 |
| 0 | 0 | 0 | 0 | 10 | 59 |
| 0 | 0 | 7 | 12 | 0 | 0 |
| 0 | 0 | 5 | 66 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 63 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 0 | 0 | 34 | 68 | 0 | 0 |
| 0 | 0 | 5 | 39 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,72,0,0,2,0,72,0,0,0,0,2,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,72,0,0,0,0,2,0,72],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[62,71,0,0,0,0,60,11,0,0,0,0,0,0,0,0,7,5,0,0,0,0,12,66,0,0,14,10,0,0,0,0,24,59,0,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,34,5,0,0,0,0,68,39,0,0,68,10,0,0,0,0,63,5,0,0] >;
D4.S3⋊C4 in GAP, Magma, Sage, TeX
D_4.S_3\rtimes C_4
% in TeX
G:=Group("D4.S3:C4"); // GroupNames label
G:=SmallGroup(192,316);
// by ID
G=gap.SmallGroup(192,316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,135,100,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=e^4=1,d^2=a^2,b*a*b=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations